There is this dairy farmer with a problem. He's tried everything he can think of, but milk production from his herd just won't budge. The dairy new online push hormones, but no .. it was time for something unconventional. He hired three consultants - a mechanical engineer, a psychologist and a physicist.

A week went by and the engineer came back with a stack of papers. The problem, according to the engineer, is the farmer's milking machines don't develop enough vacuum and the tubes to the storage tank have too many bends. A special machine is proposed, but the farmer decides to talk to the next consultant.

The psychologist takes a lot of photographs and makes some sound recordings. He compares these with those of benchmark barns and after a week reports the barn is too disorderly and the sounds from the nearby highway make things worse. The recommendation is to hire an interior decorator and subscribe to Moozak.

Nope .. time to listen to the physicist.

The physicist listens to the farmer describe the problem. "hmmm..." she says as she pulls a half used Fieldnotes pad, pencil and slide rule from her backpack. Gazing up at the sky focused on nothing in particular she says...

consider a spherical cow

You've probably heard some variation of this. I like it as it describes how a lot of physics is done. Physics and the other sciences are usually poorly taught - it's anything but facts and equations to memorize and somehow apply. Rather it's mostly play at it core - at least how you think. You strip away anything that isn't necessary and see what you can learn. If the answer doesn't make sense you try a different approach. The trick is knowing how much to strip away and what to leave in and how to mix in other questions along the way. Ideally you're left with new insight.

A regular reader told me he was curious about my thinking process and asked if I could describe it. At first I ignored the request, but then decided to give it a try with a little real time play. Using the spherical cow joke as a seed I took a walk with a very smart friend as a sanity check. This isn't a transcript, but here's roughly how it unfolded.

First let's draw a picture of a cow. Consider a spherical cow of radius 1. I don't care what the units are and for now a sphere is the simplest shape to get started with.

Now draw another sphere of radius 2 - this will be our gigantic cow. What kind of milk production can we expect. Any linear dimension will be twice as big so we can see if barn door heights and widths will work. The udder will be part of the sphere. It will be 8 times as large as the regular cow .. the volume of any part of the cow scale as the radius cubed - it doesn't matter what part of the original spherical cow I choose, as long as the part on the larger sphere has the same shape, its volume will be r^{3} or 2^{3} = 8 times as much.

Already a problem looms. What is the pressure in the udder? Pressure is just a force spread over an area (think pounds per square inch). The force in this case is the weight the udder has to support, which increases as the cube of radius. The area of a section of the sphere is 4πr^{2} times the fraction of the sphere the udder takes up. Now the udder fraction and the 4π are the same for a spherical cow of any radius. We'll be comparing the two by dividing so they'll cancel out. The pressure will be proportional (a physicist would say "goes like..") to r^{3} divided by r^{2} -- or just the radus r. Our giant cow has twice the pressure on it's udder as the standard cow. This may be a serious cow design problem and we would need to know more to make progress. But it gives a hint of another direction.

Now draw a more accurate picture of a cow and giant cow. A small sphere connected by a cylinder to the larger sphere. What about the force on the neck? The important innards of the neck are the spine and muscle. Strength goes as the cross sectional area. So if the double sized cow is a perfectly scaled version of a regular cow, it's neck will be r^{2} or 4 times stronger. It must support itself and the head, both of which will see their weight go up by the cube. The ratio of weight to strength is simply r or 2 in this case. Another design issue for giant cattle breeders.

But what of really big animals - like a dinosaur? If the head is much bigger than the neck it will be the limiting factor. Imagine something like a cow scaled up eight times in length. The head would quickly become too heavy. Something like a big dinosaur walking on four feet with a long neck would need a very small head.

Now it's clear why the dinosaurs went extinct.

Their brains were too small. This handicap meant they never developed a space program and were unable to deal with the asteroid.

Of course this is tongue in cheek, but this sort of playful approach is exactly how a lot of physics gets done. As you proceed detail and more critical questions are added. One gets to point where measurements of Nature are needed and then you see if you can predict -- if not it's just storytelling and not science.

Back to the example. It's an interesting approach to begin thinking about sports. What kind of body types are more suited to one sport or the other? People on the list range in height from something under five foot zero to six foot eight. One might consider gymnastics and the other basketball or volleyball. Neither would be a good distance runner... Soon you need to drill deeper, but the thought process stays the same. As detail enters, calculations need to be more robust. The observation and experimentation part can be very difficult to implement. LIGO - the twin observatories for measuring gravity waves are simple to describe conceptually, but making it work took over thirty years of hard work and invention of new instrumentation and techniques (some of which has had impact in engineering and medicine). This simple concept is a few minutes at a blackboard with no equations and a string of playful questions eventually led to an entirely new form of astronomy.

The approach works in many areas where there is enough information to form some kind of a question - often one that sets limits. There are times when an hour of thinking gets you within a factor of two of a difficult problem that requires huge resources to answer to any precision. This is a good way for picking paths to follow. I've been called on to do this in scenario planning style exercises.

And for fun .. what about the folklore that the breath you just took at least one molecule from Julius Caesar's last breath. A minute or two of thinking gives the answer, but it raises some really interesting questions about the atmosphere.. The first time I thought about the problem as a teenager I was struck by the notion that the atmosphere for a plant is like a salty sea and the thirsty sailor. That leads you to think about fertilizer and the enormous scale of the process - probably the primary reason for the planet being able to sustain more than two billion people. Then you think about the energy costs and other natural limiters. This style of thinking is a great exercise for anyone who likes to discover dot connections.

Clearly one doesn't do this all the time, but it's great fun and I encourage folks to try it out. When you do it with others it's not brainstorming - brainstorming doesn't work. Rather it's a form of play. It can get addictive and can be physically dangerous for those of us who can't walk and chew gum at the same time. I naturally fall into this with one of you who has saved me from Manhattan traffic on several occasions.

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